A Robust Foundation Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers

2026-05-06

AI Summary

This work addresses the challenge of accurately and robustly solving conservation law systems in the absence of explicit governing equations or partial differential equation coefficients. The authors propose a novel hypernetwork architecture that integrates a recurrent Vision Transformer with a flux neural operator (Flux NO), dynamically generating Flux NO parameters through context-aware injection to enable equation-free modeling and simulation. This approach is the first to combine recurrent Vision Transformers with flux neural operators, leveraging contextual information to generalize across unseen flux functions and previously unencountered conservation systems. Experimental results demonstrate that the proposed framework achieves high-accuracy, long-term stable predictions across diverse conservation laws, significantly outperforming existing neural operator methods in robustness, generalization capability, and adaptability to unknown fluxes.

Abstract

We propose an architecture that augments the Flux Neural Operator (Flux NO), which combines the classical finite volume method (FVM) with neural operators, with ViT-based context injection. Our model is formulated as a hypernetwork: it extracts solution dynamics over a finite temporal window, encodes them with a recurrent Vision Transformer, and generates the parameters of a context-conditioned neural operator. This enables the model to infer and solve conservation laws without explicit access to the governing equation or PDE coefficients. Experimentally, we show that the proposed method preserves the robustness, generalization ability, and long-time prediction advantages of Flux NO over standard neural operators, while delivering reliable numerical solutions across a broad range of conservative systems, including previously unseen fluxes. Our code is available at https://github.com/xx257xx/CONTEXT_FLUX_NO.

Problem

• conservation laws
• neural operators
• flux
• PDE
• numerical solution

Innovation

• Flux Neural Operator
• Vision Transformer
• Conservation Laws
• Hypernetwork
• Context Injection